Functional Analysis

This text is a self-contained introduction to the three main families that we encounter in analysis – metric spaces, normed spaces, and inner product spaces – and to the operators that transform objects in one into objects in another. With an emphasis on the fundamental properties defining the spaces, this book guides readers to a deeper understanding of analysis and an appreciation of the field as the “science of functions.”Many important topics that are rarely presented in an accessible way to undergraduate students are included, such as unconditional convergence of series, Schauder bases for Banach spaces, the dual of ℓp topological isomorphisms, the Spectral Theorem, the Baire Category Theorem, and the Uniform Boundedness Principle.The text is constructed in such a way that instructors have the option whether to include more advanced topics.Written in an appealing and accessible style, Metrics, Norms, Inner Products, and Operator Theory is suitable for independent study or as the basis for an undergraduate-level course. Instructors have several options for building a course around the text depending on the level and interests of their students.Key features:Aimed at students who have a basic knowledge of undergraduate real analysis. All of the required background material is reviewed in the first chapter.Suitable for undergraduate-level courses; no familiarity with measure theory is required.Extensive exercises complement the text and provide opportunities for learning by doing.A separate solutions manual is available for instructors via the Birkhäuser website (www.springer.com/978-3-319-65321-1). Unique text providing an undergraduate-level introduction to metrics, norms, inner products, and their associated operator theory.

This book collects the notes of the lectures given at the Advanced Course on Crossed Products, Groupoids, and Rokhlin dimension, that took place at the Centre de Recerca Matemàtica (CRM) from March 13 to March 17, 2017. The notes consist of three series of lectures. The first one was given by Dana Williams (Dartmouth College), and served as an introduction to crossed products of C*-algebras and the study of their structure. The second series of lectures was delivered by Aidan Sims (Wollongong), who gave an overview of the theory of topological groupoids (as a model for groups and group actions) and groupoid C*-algebras, with particular emphasis on the case of étale groupoids. Finally, the last series was delivered by Gábor Szabó (Copenhagen), and consisted of an introduction to Rokhlin type properties (mostly centered around the work of Hirshberg, Winter and Zacharias) with hints to the more advanced theory related to groupoids.

Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.Le Livre de Théories spectrales est consacré à l'étude des algèbres normées et de leurs applications. Le premier chapitre met en place la théorie fondamentale des algèbres de Banach et des algèbres stellaires. Nous y présentons l'équivalence de catégories entre algèbres stellaires commutatives et espaces topologiques localement compacts, ainsi que le calcul fonctionnel holomorphe en plusieurs variables dans une algèbre de Banach commutative. La transformation de Fourier, qui est l'un des outils mathématiques les plus universels, est étudiée au second chapitre, dans le cadre des groupes localement compacts commutatifs. Le texte est complété par de nombreux exercices. Ces deux chapitres forment une édition entièrement refondue de l'édition de 1967. The Elements of Mathematics of Nicolas Bourbaki have the goal of giving a rigorous and systematic presentation of mathematics starting from the foundations, without prerequisites. The book of Spectral Theories is devoted to the study of normed algebras and their applications. The first chapter establishes the basic theory of Banach algebras and C*-algebras. We present the equivalence of categories between commutative C*-algebras and locally compact topological spaces, as well as the holomorphic functional calculus in several variables in a commutative Banach algebra.The Fourier transform, which is one of the most universal mathematical tools, is studied in the second chapter, in the context of locally compact commutative topological groups.The text is accompanied by many exercices.These two chapters are completely updated new versions of the 1967 original edition.